On total transitivity of graphs

TL;DR

This work introduces and studies total transitivity, $Tr_t(G)$, defined via partitions obtained by iteratively removing total dominating sets. It provides structural results for split graphs, proving $1\le Tr_t(G)\le \omega(G)-1$ with characterizations for $Tr_t(G)=1$ and $Tr_t(G)=\omega(G)-1$, and presents necessary conditions for intermediate values. The decision version is NP-complete for bipartite graphs, while a linear-time approach is obtained for bipartite chain graphs. A polynomial-time algorithm is developed for trees, using a bottom-up computation of rooted total transitive numbers to obtain $Tr_t(T)$. Overall, the paper advances both the theoretical understanding and the algorithmic tractability of total transitivity across several graph classes, while outlining open questions for further study.

Abstract

Let $G = (V, E)$ be a graph where $V$ and $E$ are the vertex and edge sets, respectively. For two disjoint subsets $A$ and $B$ of $V$, we say that $A$ \emph{dominates} $B$ if every vertex of $B$ is adjacent to at least one vertex of $A$. A vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ is called a \emph{transitive partition} of size $k$ if $V_i$ dominates $V_j$ for all $1 \leq i < j \leq k$. In this article, we study a variation of the transitive partition, namely the \emph{total transitive partition}. The total transitivity $Tr_t(G)$ is defined as the maximum order of a vertex partition $π= \{V_1, V_2, \ldots, V_k\}$ of $G$ obtained by repeatedly removing a total dominating set from $G$ until no vertices remain. Thus, $V_1$ is a total dominating set of $G$, $V_2$ is a total dominating set of the graph $G_1 = G - V_1$, and, for $2 \leq i \leq k - 1$, $V_{i+1}$ is a total dominating set in the graph $G_i = G - \bigcup_{j=1}^i V_j$. A vertex partition of order $Tr_t(G)$ is called a $Tr_t$-partition. The \textsc{Maximum Total Transitivity Problem} is to find a total transitive partition of a given graph with the maximum number of parts. First, we characterize split graphs with total transitivity equal to $1$ and $ω(G) - 1$. Moreover, for a split graph $G$ and $1 \leq p \leq ω(G) - 1$, we provide necessary conditions for $Tr_t(G) = p$. Furthermore, we show that the decision version of this problem is NP-complete for bipartite graphs. On the positive side, we prove that this problem can be solved in linear time for bipartite chain graphs. Finally, we design a polynomial-time algorithm to solve the \textsc{Maximum Total Transitivity Problem} in trees.

Figures (6)

• Figure 1: Example of a split graph where $Tr_t(G)=1$ and $Tr(G)=\omega(G)+1=6$
• Figure 2: Total transitive tree of order up to $3$ along with its unique total transitive partition
• Figure 3: Example of a split graph
• Figure 4: Construction of $G'$ from $G$ where $m=3$ and $T$ represents a $\mathcal{T}_3$
• Figure 5: Partition of $\mathcal{T}_3$

Theorems & Definitions (48)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Proposition 3
• Proposition 4
• Proposition 5
• proof
• Proposition 6